Because angle APB= angle PAD
So corner pad = corner EPC
Thus, △APD is similar to △PCE in two cases.
The sine of angle c is 4/5.
Do DH perpendicular to BC and H after D.
Then CH=3, DH=4, BH=5, BC=8.
Let BP be x, then
1, angle ADP= angle C.
From the cosine value of the angle c, we can see that the angle ADP is an acute angle at this time, so P is on BH.
So ph = BH-x = 5-X.
According to the sine theorem, we know that the area of a triangle is
1/2×AD×DP×sin angle ADP
= 1/2 × 5 × PD×4/5
=2×PD
In addition, the bottom of the triangular row ADP is AD=5.
The height and DH are equal to 4.
So the area of the triangle ADP is also equal to
5×4/2 = 10
So you get the equation.
2×PD= 10 = >PD=5
Because PD= under the root sign (the square of PH+the square of DH)
DH=4，
So PH=3
So x=5-PH = 2.
Test: When BP=2
It is found that point e will be on the extension line of CD, not on CD.
So this situation does not meet the meaning of the question.
This is the first case.
2, angle APD= angle C.
Pay attention at this time,
Test a situation:
What happens if p and h coincide?
If p and h coincide, we can know the tg angle APD=AD/PD=AD/HD=5/4.
Tg angle C=4/3.
It shows that when they overlap, the angle APD is smaller than the angle C.
Because p moves from left to right,
The angle APD has been decreasing,
This means that point p is still on BH at this time,
Not on HC, although this discussion
It doesn't matter if you don't write, but it is still more important.
Then, continue to calculate:
The ADP area of the triangle is still expressed by base × height /2, that is, 10.
At the same time, it is expressed by sine theorem as follows
1/2 × AP × PD × sin angle APD
= 1/2 × sin angle C × AP × PD
=2/5 × AP × PD = 10
So AP×PD = 25
According to the cosine theorem, there are
Adsquare = apsquare+pdsquare-2cos angle APD×AP×PD.
=AP squared +PD squared -2×AP×PD×3/5
Substituting AP×PD=25, there are
AD square =AP square +PD square -30
therefore
25 = square of AP+square of PD -30
AP square and PD square can be expressed by Pythagorean theorem in BP and PH respectively.
PH=5-BP。
So we can get an equation about BP:
25 = square of BP+16+square of ph+16-30.
therefore
Blood pressure squared +PH squared =23
BP squared +(5-BP) squared =27
Solve the equation with BP=(5+ root number 2 1)/2.
Or BP=(5- radical 2 1)/2.
Check the values of two BP.
All match 0.
But when BP=(5- radical number 2 1)/2, because BP is too small (about equal to 0.2 1), it will make
E is not on the CD, but on the extension line of the CD, which does not meet the meaning of the question and is discarded.
So BP=(5+ radical number 2 1)/2.
Answer over.
The level is limited, please correct me.